Ss. Ralescu et Ml. Puri, WEAK-CONVERGENCE OF SEQUENCES OF FIRST PASSAGE PROCESSES AND APPLICATIONS, Stochastic processes and their applications, 62(2), 1996, pp. 327-345
Suppose {X(n)}(n greater than or equal to 1) are stochastic processes
all of whose paths are nonnegative and lie in the space of right conti
nuous functions with finite left limits. Moreover, assume that X(n) (p
roperly normalized) converges weakly to a process X, i.e., for some de
terministic function mu and theta(n)-->0, theta(n)(-1)(X(n)-mu)-->(d)X
. This paper considers the description of the weak limiting behavior o
f the sequence of first passage processes <(X)over tilde (-1)(n)>(t)=i
nf{x:<(X)over tilde (n)>(x)greater than or equal to t} where <(X)over
tilde (n)>(x)=rho(nx)X(n)(x) and rho(.) is such that (X) over tilden$(
x) has nondecreasing paths. We present a number of important motivatin
g examples including empirical processes associated with U-statistics,
empirical excursions above a given barrier, stopping rules in renewal
theory and weak convergence in extreme value theory and point out the
wide applicability of our result. Weak functional limit theorems for
general quantile-type processes are derived. In addition, we investiga
te the asymptotic behavior of integrated kernel quantiles and establis
h: (i) an invariance principle; (ii) a strong law of large numbers; an
d (iii) a Bahadur-type representation which has many consequences, amo
ng which is a law of the iterated logarithm.